We take it for granted. We understand it. It is obvious what temperature is. Cold, warm, hot…obvious.

But how many of us have asked the next question: what is the real difference between a hot stone and a cold one? The answer is interesting and helps us to realise that measuring temperature is much trickier than we tend to suppose.

Over many hundreds of years, many clever people have devised lots of experiments to understand what temperature is, I hope in this article to round up the facts!

Temperature and Energy

For much of history, there were only a few sources of heat – the sun, fire, lava and of course the warmth of living creatures.

People were puzzled by what created it, but it was immediately obvious that it had one consistency – whenever it had the chance, it flowed – put something hot next to something cold, and the heat would flow.

Of course you could argue that it was the ‘cold’ that flowed (the other way), but there were no obvious sources of ‘cold’. While ice was clearly cold, it was not a sustainable ‘source’ of cold the way a fire was.

It was also noted that heat melted things – like fat or butter and that it make some liquids (like molasses) thinner. It could even boil water and make it ‘vanish’. The mechanisms for these were unknown and a source of fascination for early scientists.

Early experimenters noticed that gases would increase in volume upon heating, and that compressing gases would cause them to heat up. They also investigated other sources of heat, like friction, (rubbing your hands together).

It was the work with gas that led to the big breakthrough. Boyle and Hooke, as well as Edme Marriotte, working in the 17th century, realized that the temperature of a gas would increase consistently with pressure, and like-wise, decrease consistently with pressure. This sounds unremarkable, until you note that you can only decrease pressure so much…

Once you have a vacuum (no pressure), you should have ‘no temperature’. Thus their observations implied that there really was a limit to how cold things could get, and predicted it was around -275 Celsius. They were of course unable to cool anything that far simply by expanding it because heat always flows into cold things, so to achieve this you need much better insulation than they had available.

So they had a big clue in the search to understand what temperature is, but still no explanation.

It took until 1738 until another great scientist moved us forward. Daniel Bernoulli realised you could use Newton’s (relatively new) laws to derive Boyle’s temperature-pressure relationship. He basically asked: what if gas was made of a large number of very small billiard balls flying around crashing into everything? What if pressure was just the result of all these collisions? Using this theory he realised, for the first time I think, what temperature truly is.

Source: Wikimedia Commons

It turns out that his model equated temperature with the speed of the billiard balls. A hot gas only differs from a cold gas in the speed of the molecules flying around. Faster molecules crash with more momentum and thus impart more pressure. Squashing the gas into a smaller volume does not give them more speed, but means more collisions each second, so higher pressure.

This is a pretty serious finding. It basically says ‘there is no such thing as temperature’. There is only lots of little balls flying around, and their number and speed dictate the pressure they exert, and there is no ‘temperature’.

If we put a thermometer into the gas, what is it detecting then? Great question.

It turns out that solids are also made of lots of balls, except, instead of being free to fly around, they are trapped in a matrix. When a solid is exposed to a hot gas, it is bombarded by fast flying atoms. When a solid atom is hit, instead of flying off, it starts to vibrate, like a ball constrained by a network of springs.

So the ‘temperature of a solid is also a measure of speed of motion, but rather than linear speed it’s a measure of the speed of vibration. This makes a lot of sense – as the solid gets hotter, the balls are going literally ‘ballistic’ and eventually have enough speed to break the shackles of the matrix (aka melting).

Source: Wikimedia Commons

So this model of heat as ‘movement’ not only explains how gases exert pressure, but also explains how heat flows (through molecular collisions) and why things melt or vaporise.

More importantly, it shows that temperature is really just a symptom of another, more familiar, sort of energy – movement (or kinetic) energy.

Energy is a whole story of its own, but we can see now how energy and temperature relate – and how we can use energy to make things hot and cold.

Making Things Hot

There are many easy ways to make things hot. Electricity is a very convenient tool for heating – it turns out that when electric current flows, the torrent of electrons cannot help but buffet the atoms in the wire, and as they are not free to fly away, they just vibrate ever faster, ‘heating’ up.

Another way to heat things is with fire. Fire is just a chemical reaction – many types of molecules (like methane, or alcohol) contain a lot of ‘tension’, that is to say, they are like loaded springs just waiting to go off. Other molecules (often oxygen) hold the ‘key’ to unlocking the spring, and when the springs go off, as you can imagine, it is like a room full of mousetraps and ping-pong balls – and all that motion – means heat.

Making Things Cold

Manipulating energy flows to make things cold is much trickier.

One way it to just put the thing you want to cool in a cold environment – like the north pole. But what if you want to make something colder than its surroundings?

Well there is a way. We learned earlier that gases get hot when compressed – it turns out they do the opposite when decompressed or ‘vented’. This is the principle that makes the spray from aerosol cans (deodorant, lighter fluid, etc) cold. So how can we use this? First we use a compressor to compress a gas (most any gas will do); in the process it will warm up, then you let it cool down by contacting it with ambient air (through a long thin copper tube, but keeping it compressed), then decompress it again – hey presto, it is cold! Pump this cold gas through another copper tube, inside a box, and it will cool the air in the box – and hey presto, you have a refrigerator.

Measuring Temperature

Before we had thermometers, temperature was generally estimated by touch.

However this is where temperature gets tricky. Because the temperature we feel, when we put our hand on the roof of a car is not really the temperature of the car, it’s really the measure of energy flow (into our hand), which relates to the temperature, but also relates to the conductivity of the car.

This is why hot metal feels hotter than hot wood, why cold metal feels colder than cold wood – the metal, if at a different temperature to your hand, is able to move more heat into you (or take more heat away) faster than wood can. Thus our sense of temperature is easily fooled.

The ‘wind-chill factor’ is another way we are fooled – we generally walk around with cloths on, and even without clothes we have some body hair – therefore, we usually carry a thin layer of air around with us that is nearly the same temperature as we are. This helps us when it is cold and when it is hot – however, when the wind blows it rips this layer up and supplies fresh air to our skin – making us feel the temperature more than usual. Also, because our skin can be damp, there can be evaporative effects which can actually cool you below the air temperature.

Scientists have long known that we cannot trust ourselves to measure temperature, so over the ages many tricks have been developed – can the object boil water? Can it freeze water? A long list of milestone temperatures was developed and essential knowledge for early scientists – until the development of the lowly thermometer.

It was noted that, like gases, solids and liquids also expand upon heating. This makes intuitive sense if you think of hot molecules as violently vibrating – they push one another away, or at least if the charge (electric charge is what holds these things together) is spread just a little thinner, adjacent molecules will have slightly weaker bonds.

The expansion of liquids may only be very slight, and if you have a big volume of liquid in a cup, the height in the cup will change only very slightly, but if its in a bottle with a narrow neck, the small extra volume makes a bigger difference to the level. This principle is used in a thermometer – it’s just a bottle with a very narrow and long neck. The bigger the volume and the narrower the neck, the more sensitive the thermometer. Of course the glass also expands, so it is important to calibrate the thermometer – put it in ice water, mark the liquid level – then put it in boiling water and mark the new level. Then divide the distance between these marks into 100 divisions – and hey presto! you have a thermometer calibrated to the centi (hundred) grade (aka Celsius) scale. Now you know where that came from!

Every time I hear another pundit explaining their theory behind England’s failure at the world cup I get all hot and bothered. My wife could literally not care less about football or my feelings on the subject, so I thought I would share my them with you

You see, it seems that football does not benefit, as cricket and baseball do, from that important type of pundit, the statistician.

For if they did, they would realise that failing to make it into the last 16 this year is not a failure at all.

Why? Because we have to remember that there are, at time of writing, 209 national men’s teams registered with FIFA – and that FIFA estimates that 250 million people play the beautiful game. So just getting to the finals is a real achievement.

On the other hand you can argue that England should be in the top flight – it has a decent population, it has money to spend and many aspirational heroes.

Well.. it does do well, currently, England is ranked 10th in the world, and has often been higher. But does that mean it should always reach the last 16 of the world cup? No.

Take a step back. Even with so many teams, that world ranking should mean a team like England should make it ‘usually’, but certainly not always. Indeed, it has made the last 16 every time since 1958, indeed it’s a surprise to me that they have run so long without missing out.

It has always been known that football has a ‘luck factor’ indeed this is one of its best features – upsets happen – and that is why the league has a round-robin design, and also why some tournaments are done by knockout – the league aims to find the best teams, tournaments aim to find the best moments.

England did not play badly this world cup. The goals conceded were really pretty darn good, and England had more shots on goal than their opponents. But for the rub of the green, they could have been through and lauded by all. So how can the pundits have such strong opinions?

Easy. Because it’s their job to sound like they know.

Addendum
So how have teams like Brazil, Spain, Germany, Italy, France and the Netherlands managed to spend so much time in the top 3? If it was all luck, they would not. Well, this means that there are actually recipes for better performance, elusive but real…

However, just as the government would hate for you realise they do not control the economy, football pundits and administrators alike would not like you to know (or indeed to know themselves) that this recipe is largely outside of their control – the biggest factors being: population size, other games to play, weather, virtuous circles (inspiration, promise of fame, etc) and last but certainly not least, luck.

Well no wonder there’s a sudden surge of doubt about vaccination – not because we’ve realise how well it works, but because we’ve forgotten what life was like before vaccines.

I am a parent. I can understand the idea of injecting my child with something that turns out to be harmful would be hard to bear. First do no harm, they say. Can we be sure there is no harm?

Sounds a like a fair question… but… isn’t. It’s not that simple.

Having a vaccine may seem like a dangerous intervention, and surely no action is better than a risky one?

Well if you believe that, you are not alone, but are making a cognitive error, the error to assume inaction is a virtue. As Haile Selassie said “Throughout history, it has been the inaction of those who could have acted; the indifference of those who should have known better; the silence of the voice of justice when it mattered most; that has made it possible for evil to triumph.”

In order to help those stuck in this cognitive trap, I would ask them to consider this thought experiment.

Rather than choosing to go to a doctor for a vaccine, say the famous ‘MMR’ triple-jab, or choosing to stay at home, let’s imagine the choice was one of two medications, one the MMR, with its supposed* risk of autism, but which gives immunity to measles, mumps and rubella – and the other is a compound with no known benefits, but a known risk of causing measles, mumps and rubella. And of course the added possibility of causing an epidemic at the same time.

(*) Let’s add to the mix the fact that there is zero evidence of any link between MMR and autism, and also point out that failing to immunise your kids is reckless cruelty not only to them, but also to all those who cannot be vaccinated for ‘real’ reasons, such as being too poor, too young or too sick.

To those still on the fence, this last bit is for you.

Do you not realise that what seems like concerned parenting has actually cost real people their lives? For what?

Yes, drugs go wrong. Yes, corporations are often tempted to hide bad news. Yes, the human body is complex and we do not understand every last detail.

But if you think that those pressures overwhelm the ‘good’ in mankind that has led life expectancy to climb gloriously every single decade this last century, then you are missing all the good news.

No small part of the doubling in life expectancy was due to vaccines.

Please don’t take me with you back to 1900. Rather look at this picture of what smallpox does and thank your lucky stars she’s not your child.

I have been keeping my eye on the evidence that ‘life experience’ can be passed on in your genes.

It has been proposed many times as a mechanism of evolution, and indeed was considered likely before the concept of ‘selection’ was understood. It’s attractive because saying that ‘survival’ is the one and only way to adding value to the genes seems, well, wasteful.

Surely, you’d think that a fear of snakes based purely on the idea that people who were not afraid of snakes were ‘taken out’ of the gene pool by snakes, is less efficient than a mechanism that captures experience – that snake killed my dog, I should avoid snakes, and so should my kids…

However, once DNA was understood and shown to be dense with what seemed to be all information that could ever be needed, dissent waned to an all time low. The mechanisms of DNA were pretty clear – your DNA was set at birth and while it might mutate a little randomly, had no way to ‘learn’ from your life before being combined with a partner’s to create offspring. The case seemed pretty settled.

There remained niggles though. I worried about the speed of evolution as we have so very much to learn and so little time to evolve! So I looked for ways evolution could amp up its power. It seemed to me that nature, so darn clever at self-optimisation would make improvements to our design based on non-fatal experience, or indeed passive observation.

Others similarly concerned continued, often in the face of deep scepticism, to study what is called epigenetics, the science of heritance not coded by DNA – and thus having the potential to be edited during our lives.

I first heard about it around 10 years ago when media reported that the actual DNA sequence was not the only way info could be stored in cells – in theory the histones present can affect how the DNA ‘works’ (how genes are ‘expressed’) and their presence could thus change the characteristics of any lifeform coded therein. The type and number of histones may be few (relative to the number of base pairs in DNA) however, the many locations and orientations they can take create a fair number of possible combinations and permutations.

When I heard about this theory, I was put into a state of high curiosity. On the one hand, it was a little blasphemous, but on the other hand tantalising. If nature could find a way to combine the power of selection with the potential benefits of life experience, we could get much faster and more effective evolution.

My curiosity was soon rewarded with another possible mechanism for smuggling info to the next generation. DNA methylation – the idea is that DNA can host little ‘attachments’ in certain places. These may be temporary, and reversible, but they have now been clearly shown to alter how the DNA expresses itself.

On the face of it, the evidence that DNA expression is environment dependent is rather strong, but the idea that the environment around the DNA coil actually contains consistent and persistent intelligence picked up during our lives is much harder to prove.

And so teams have been beavering away trying to get to the bottom of this, and this week one such group has fresh news for us. A Nature Neuroscience paper has tested the theory in a rather clever way.

They started by teaching some mice a new fact (that a certain smell would be associated with trauma) and then later, tested their kids. Lo and behold the kids whose parents had been taught fared significantly better in the test than untrained, unrelated mice.

This may sound a little trivial, but you must remember that the current ‘popular’ understanding of genes is that they only gain intelligence by surviving – or more precisely they shed stupidity by dying. However, here we are seeing information pass between generations without the need for anyone to die.

Furthermore, because of the carefully selected lesson taught to these mice, the researchers were actually able to see that a specific part of the DNA, while not different in design was nevertheless more active.

Now, I do not know enough about DNA to double-check this claim, but you can rest assured others will – because that’s what journals are for – and in this case the implications are huge.

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Like what, you ask?

Well, off the top of my head, it means that much of what we do between birth and reproduction will affect all our descendents – this undermines the idea that one’s body is one’s own to do with as one pleases.

It also indicates that there is potential for us to deliberately control the expression of our DNA, allowing us to do some genetic engineering without actually changing the DNA sequence.

More importantly, and more controversially, it would mean natural selection would not need to explain all the marvellous diversity we see around us on its own.

It remains to be seen what proportion of our ‘design’ is coded for outside the DNA, or indeed how much this mechanism can improve or speed up evolution, however I for one hope it works out to be right and that mother nature has indeed figured out how to seriously boost the power of selection.

I read some comments on Scientific American today that instantly made my blood boil. Or cavitate at least.

It was an explanation of how tall trees get water right up top. No I never thought about that before either.

Anyhow, anyone who’s drilled a borehole knows you can only suck water up 33 ft before you get a vacuum forming, water boiling and general pumping failure. Hence the need to put a pump at the bottom of a deep borehole.

Now, I had always thought capillary action was what sucked water up plants, handily bypassing this issue, and there, right in the comments, it was asserted that this was a ‘common misconception’…

What, me wrong!? Never!

After the shock, I did what a good scientist is supposed to do, fighting the desire to simply namecall, I watched her darn video.

I remained skeptical. Very skeptical. I again overcame desires to write rude comments in youtube and went a read up on it properly…

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Ok, so it turns out that there is some sort of truth to it: indeed some clever people believe water can be ‘sucked’ to the top of tall trees, which does indeed require negative pressure.

So I ask, why won’t the water boil?

Because, they say, it’s ‘meta-stable’. Like super-cooled water, or superheated water, water can supposed go to ‘tension’ without boiling if only you can prevent that initial bubble forming. Simple!

A little more thinking and internal wrangling, and I slowly conceded it just might be. Yes, ok, negative pressure is not really all that radical, it is essentially tension. It’s common in solids, it’s just the idea that water can be ‘tense’ that is difficult to get one’s head around.

So, the process had begun; I started to consider that maybe I was wrong. It’s not pleasant folks, and I am not trying to beat my own drum, I am sure there are plenty of other times when I’ve failed this test, it was just interesting because here I think I passed it…

Anyway, back to the point. Alas, I then read even more deeply, that though I find myself agreeing that water can indeed be under tension, and that sort of does mean negative pressure, I’ve yet to be convinced that ‘wicking’ it not at least involved in tree sustenance. Anyone who has dropped a dry cloth in water knows the water climbs into the fabric.

Furthermore, if there was negative pressure in the tree’s ‘pipes’ why wouldn’t they collapse?

It took deeper digging, but now all my cognitive dissonance is resolved, and I feel just fine by closing my investigation with this makeshift conclusion: that while trees do suck water up (via transpiration and the pull of surface tension in narrow openings) the pressures needed are not too crazy BECAUSE OF THOSE GOOL OL’ WICKING EFFECTS!!

Yup, I have to conclude that the attraction of the fluid for the xylem walls helps ‘keep the water up’ and thus preventing it from pulling too hard on the water above it.

It turns out this is what many others think [great minds for sure], and some [’nuff respect] took the steps of building a pressure probe small enough to poke into a plant’s pipework. What they found supports my newly cherished (but alas already 120-year-old) Cohesion-Tension theory of tree hydration.

In other words, while wicking (capillary action) is not a sole actor, it is there in a critical supporting role. Aaah that’s better, as you can see I wasn’t totally wrong

PS. On the other hand, negative pressures seem to be a new and reproducible fact for us to worry about!

Ok, if you’re tired of being lectured about your sweet tooth or laziness (or both), and just want the straight dope from an engineer, you’ve come to the right place.

You see, over the next couple of minutes, we’ll see that a human body is not much different to, say, a tractor. It’s a tough machine and just like a tractor has very few needs – a little fuel, a little air and a little water.

Ok, ok, we’re a little more complex, but when push comes to shove, we are pretty similar, let me show you…

Food = Fuel

My wife despairs, but she I must point out that she chose to marry an engineer with hardly any niceties. Yes, ok, food is more than just fuel, it’s one of the joys of life yadda-yadda, but, to the engineer in me food is just a handy Continue reading →

Yet bacteria get around, and indeed are remarkably successful. Same with viruses. So thriving as a species does not require planning, studying, concentration and imagination, all the things we humans are so arrogant about.

And I’m not just talking about surviving, I’m talking about achieving the incredible. Think of the fungi that take control of ants, get them to climb up to a good spot and hunker down so that when the fungus bursts out from the corpse like a slow motion firework (see the picture) it’s got a fair chance of spreading its spores.

Did the fungus plan it? Of course not, the trick ‘evolved’ as the most successful of many different permutations, via, of course the process of natural selection.

So what?

Well, what about humans? We like to think we are the pinnacle of evolution with our big brains and our consciousness and our self-awareness. Our abilities to plan, co-operate and imagine have led us to dominate the planet. Or have they?

Could it be, that just as no ant envisions the design of the anthill, none of us can claim to have masterminded very much? Yes perhaps a building, a harbour or a town’s zoning, but who can claim to have masterminded New York or world trade or democracy?

Surely these ‘real’ achievements are not ours to claim, but should also be laid at the door of the power of evolution, of the unstoppable force of trial and error, of the natural emergence of order from the chaos?

I have been reading some pretty strange stuff about Gravity recently. It seems there are some pretty odd folk out there who have taken thinking about physics to a new, possibly unhealthy, level.

Basically, they are crackpots. Well I guess it’s a slippery slope – one day you wonder why the earth is sucking down on you, the next you decide to spend some time on the knotty question. Soon enough you think you’ve got it, it is clearly that the earth is absorbing space which is constantly rushing down around us dragging us with it. Or similar.

So yes, its true, Einstein did not ‘solve’ Gravity, and there is still fame and fortune to be had in thinking about gravity, so this is the example I shall use today.

The trouble with Gravity is that Einstein’s explanation is just so cool! He explained that mass warps space and that the feeling of being pulled is simply a side effect of being in warped space. It sounds so outlandish, but also so simple, that it has clearly encouraged many ‘interesting’ people to have a crack at doing a better job themselves.

So, as a service to all those wannabe physics icons, I offer today a service, in the form of a checklist – what hoops will your new scientific theory have to jump through to get my attention, and that of the so-called ivory tower elite in the scientific community?

Requirement 1: Your theory needs to be well presented

Yes, it may sound elitist to say, but your documentation/website/paper/video should have good grammar. Yes, yes, one should not use the quality of one’s english to judge the quality of one’s theory, and I know prejudice is hard to overcome, but this is not my point. My point is that in order to understand a complicated thing like a physics theory it needs to be unambiguous. It needs to be clear. It needs to use the same jargon the so called ‘elite’ community uses. Invented acronyms, especially those with your own initials in them, are out.

Requirement 2: Your proposal needs to be respectful

Image courtesy of Wikimedia Commons

Again, this is not about making you bow to your superiors in the academic world. Indeed in the case of Gravity, the physics community is one of the most humble out there. While I agree academia is up it’s arse most of the time, this is about convincing the reader that you know your stuff. In order to do that, you need to show that you know ‘their stuff’ too. If you have headings like “Einstein’s Big Mistake” it is a bit like saying to the reader ‘you are all FOOLS!’ and cackling madly. Don’t do it!

Respect also means you need to answer questions ‘properly’. That means clearly, fully, and in the common language of the community. You cannot say “its the responsibility of the community to test your theory”. This is a great way to piss people right off. It is your responsibility to make them want to. This usually means dealing with their doubts head-on, and if you can do that, I promise you they will then want to know more.

Requirement 3: You need to develop credibility

Sorry, as you can see we have yet to consider the actual merit of the theory itself. I wish it were not so, but we are humans first and scientists second. We cannot focus our thoughts on a theory if we doubt the payback. And if you say that aliens came and told you the scientific theory, then people are unlikely to keep listening, unless, perhaps they’re from Hollywood.

But seriously, credibility is the hidden currency of the world, it opens doors, bends ears and gets funds. It takes professionals decades to build and it is really rather naive to waltz into a specialism and expect everyone to drop their tools and listen to you.

That said, the science world is full of incomers, it is not a closed shop as some would you believe. If you follow requirements 1 and 2, and are persistent (and your theory actually holds water) then you are very likely to succeed.

Now this is rather remarkable. Can it really be that you can calculate the speed of light to 9 significant figures from just the earth’s gravitational acceleration and the length of a year? Intuitively I suspect you could (eventually), but then I started to think, well, what if the earth was irregularly shaped? The gravitational constant is actually not all that consistent depending on where you are either. So I checked, then I noticed he said ‘lunar year’. What? Why? What is a lunar year? Then I calculated that the time he used was 354 days, which isn’t even a lunar year. Add to that that he gives the acceleration of gravity on earth to 9-figures despite the fact that nobody knows it that well (like I said it is location dependent). Does he do the same test for other planets? No. Well what if they have no moon!

Requirement 5: The theory needs to be be consistent with well-known observations

Now if your theory has got past requirements 1-4 , well done to you, you will be welcome to join my table any time. Now is when you may need some more help.

Once a theory is consistent with itself, it now needs to agree with what we see around us. It needs to explain apples falling, moons orbiting, light bending and time dilating. This is the hardest part.

It cannot leave any out, or predict something contrary to the facts. It cannot be vague or wishy-washy. It needs the type of certainty we only get from the application of formal logic – and that old chestnut – mathematics.

No you cannot get away without it, there is no substitute for an equation. Equations derived using logic take all the emotion out of a debate. And they set you up perfectly for requirement #5.

Requirement 6: The theory needs to make testable predictions

If your theory has got past the 5 above, very nice job, I hope to meet you one day.

We are all set, we have a hypothesis and we can’t break it. It has been passed to others, some dismiss it, others are not so sure. How do you create consensus?

Simple, make an impressive prediction, and then test that.

Einsteins field equations for example, boldly provide a ‘shape’ of space (spacetime actually) for any given distribution of mass. With that shape in hand you should then be able to predict the path of light beams past stars or galaxies. These equation claimed to replace Newton’s simple inverse square law, but include the effects of time which creates strange effects (like frame dragging), which, importantly could be, and were, tested.

The beauty of these equations, derived via logical inference from how the speed of light seems invariate, and now proven many times, is that they moved physics forward. Rather than asking, ‘what is gravity’, the question is now ‘why does mass warp space’. It’s a better question because answering it will probably have implications far beyond gravity – it will inform cosmology and quantum theory too.

Conclusion

So if you are thinking of sharing with the world at last your immensely important insights, and want to be listened to, please remember my advice when you are famous and put in a good word for me in Stockholm. But please, if, when trying to explain yourself, and are finding it tough, please please consider the possibility that you are just plain wrong…

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Jarrod Hart is a practicing scientist, and wrote this to shamelessly enhance his reputation in case he ever needs to peddle you a strange theory.

Yes, we are pretty awful, and it’s pretty amazing we manage to get through the day. The reason we do is that our brains were not designed to remember long numbers or to calculate square roots, we were designed to …get through the day.

Thus it’s no surprise that we can spot tigers hiding in the shrubbery, and judge someone’s intent from the curl in the corner of their mouth – things computers can’t even dream of!

Amazing Things the Brain Can Do

There are some really remarkable abilities the evolutionary arms race has given us. Consider for a moment how hard it is to teach these skills to a computer:

Theory of mind – our ability to realize that others have motives and intentions and the ability to guess them reasonably well

Inventiveness – our ability to make connections from disparate fields

Much has been said about these skills, and in particular, much value has been placed on theories about our inventiveness – if only we can understand how we invent, we can unleash a torrent of innovation!

The ideas usually run something like this: the human mind is so highly integrated that many concepts are forced to overlay one another so connections are inevitable – while others suggest the mind reviews new learning each night during sleep and tries to spot patterns, suggesting our innovative spark is really just our pattern recognition skill in disguise [1].

While I suspect there is truth to both theories, there is probably more to it than that…

Another Amazing Skill Often Overlooked

Now – if you have ever caught a child being naughty, you may have been lucky enough to see another remarkable human talent…

Lying.

Lying is tricky. Lying requires amazing computation – it needs theory of mind, it requires creativity, and does its invention under pressure.

Lying requires creating an entire alternate reality that fits the evidence but makes you look innocent of all crimes! It’s so hard that young kids don’t always get it quite right, but at some point most of us master the art. Our brains can also be switched to this mode of inventive overdrive in another way: when we attempt to explain incomplete data.

The most common opportunity to fit a narrative to incomplete data is when we recall faded memories – it turns out many of us can bring out our internal Dr. Seuss when recounting our roles in past events.

And because we all like to think of ourselves as pretty darn awesome, our memories cannot contain any information that could contradict this most evident truth. Thus when we recall situations when we did something downright shameful, our brains become positively electrified and we will magic up perfectly good reasons for what we did out of thin air.

Almost everyone can do it. However, if you ask us to write a short bit of utter fiction, our ability instantly vanishes.

Leveraging Brain Power

So the question is this… how can we tap into these remarkable abilities? Do creative people already do it?

To most people, maths is just something we learned in order to avoid being ripped off. To some, maths is an essential tool, helpful in modelling plague outbreaks or cracking encryption ciphers.

However, for an elite few, maths is simply a parallel universe and they are its explorers.

Today let us discuss what I consider perhaps the most beautiful discovery to date. But first, some introductions…

Part 1: Consider, to start, the circle

If you have a wheel a metre across, it will roll out about 3.14159…metres each revolution. This number, which we call π turns out to be some sort of fundamental property of ‘space’.

The Greeks were not very happy about the ‘messiness’ of this number. They preferred numbers that could be expressed as fractions – while 22/7 was close to π, it was not exact and they lost a lot of sleep trying to find a neat way to write π.

Mathematicians have since grudgingly accepted that it cannot be written as a fraction, and indeed it cannot be written down at all because it has no ‘pattern’ and never ends, those digits just keep coming at (almost) random! Here are the first 100…

Understandably, they decided to call this sort of number ‘irrational‘.

Part 2: Consider now, ‘powers’

Mathematicians may work tirelessly on some very pointless looking things, however, they are still fairly lazy when it comes to writing stuff down. They like shorthand. So rather than writing 3+3+3+3+3 they invented ‘multiplication’, giving them 5×3.

Likewise, rather than writing 3x3x3x3x3 they invented ‘powers’, so they could write 35.

Of course they then realized these tricks could be extended past ‘whole’ numbers. 2.5×3 is 7.5. But what about 32.5?

It works of course, the answer turns out to be about 15.59 plus change.

But what does it mean? 32.5 is three, times by itself, 2.5 times! or 3x3x30.5. What on earth is that?

Well it turns out, when you ponder this (maybe I should say, if you ponder this), that 30.5 is the same as √3. So ‘root three’ is three times itself half a time…

[pregnant pause]

Ok, let’s look at it another way

Consider, for example 32x33, which is the same as (3×3)x(3x3x3) which is the same as 3x3x3x3x3 which is the same as 35 , so 3(2+3).

So using that logic…

3 = 31 = 3(0.5+0.5) = 30.5 x 30.5

And what times itself is equal to 3? Well √3! So 30.5 is √3…

It makes sense now, and we can even get used to saying things like 31.9 x 30.1 = 9.

Of course, these fractional powers also commonly yield those ‘messy numbers’, so abhorred by the Greeks. √3 is, roughly:

1.73205080756887729352744634150587236694280525381038062805580…

The logic follows through for negative numbers. 3-2 is just 1/32 which is 1/9.

Part 3. Now consider ‘e’

y=e^x. The slope is always the same as the value! This has the interesting effect that the tangent to the line always intercepts the y axis precisely 1 unit back…

y=e^x

Here is a third sort of messy number, one which the Greeks are probably glad they missed. We have Leonhard Euler to thank for discovering this one.

He noted there was a number ‘e‘ giving an equation of the form y=ex (see the graphs pictured), where the slope of the curve is the same as the height of the curve at each point.

Strange and pointless sounding perhaps but pretty simple. So y=2x doesn’t work, y=3x doesn’t work, but by trial and error you can find a value for a that works, which is, roughly:

Now solve for x. Seems pretty easy, but really you are cheating. There is no number that solves that equation. Really, to solve it you had to ‘invent’ the concept of a negative number.

Ok. Now consider the equation x2 + 1 = 0

Ah. Trickier! However it turns out that we can do the same trick; this time we simply invent another sort of number – the ‘imaginary’ number. Now if you’ve never heard about these numbers before, you may think I’m joking. Alas I am not. This is what mathematicians have been up to for the last few hundred years, just making stuff up as they go along.

So we define i as √-1, or a number, that when multiplied by itself, yields the more respectable -1.

Aside: Just as -1 is a number which, when multiplied by any negative number renders it decent (i.e. positive) once more.

So i2 + 1 = 0 and the equation is solved. It turns out mathematicians were suddenly able to solve loads of really tedious equations using this trick, which made their entire week.

So we have now got i! At last we are ready to put the puzzle together.

Simplicity emerges from the complex…

So, now I ask, what happens if you raise e to the power of i? What does it even mean? It means, e times itself √-1 times. Ouch. Nonsense surely?

Well it works out at roughly 0.540302306 + 0.841470985 i, which is a right mess, something they call, for fairly self-explanatory reasons, a complex number.

So now lets stick our old friend from the circle, π, in there and see what happens:

What is eiπ?

Surely an even bigger mess? I mean its all these messy irrational numbers combined with this home-made imaginary number…

Today we heard that a group of seismologists have been jailed over the advice they gave prior to the 2009 quake in L’Aquila in Italy which killed over 300 people.

Is this good and right?

Well let’s start by imagining you lived in l’Aquila and a series of tremors had the town worried. A group of experts comes to town, and after a long meeting, come out and declare the risk is miminal. So rather than evacuating, you go home, rest assured, only to lose your family two days later.

I would want blood. I would want to lash out. My family could have been saved!

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So at first glance this judgement may seem like a last sad ripple from a tragic event, however, it isn’t. Upon further reflection, this judgement has serious implications for the relationship between science and the law.

I have always been troubled by how a guilty verdict means ‘you did it’ even if you didn’t, turning ‘beyond a reasonable doubt’ into no doubt at all. I have previously blogged about pragmatism in law, and how you cannot be 80% guilty and thus serve 80% of a sentence.

With such pressures on the law to come to a clear conclusion, it was only a matter of time before a court was asked to decide on whether scientific advice was correct…

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So today, when a court decided that scientists were ‘too reassuring’ to the public, my alarm bells were set ringing and still haven’t stopped.

Armed with the hindsight that an earthquake in fact did occur, it is natural that the families of the victims are appalled by the advice they got. However, in this case it is critical to also put yourself into the shoes of the scientist – in the days before the quake.

They must have honestly doubted a quake would happen. If you had expected a quake you would surely have said so!

In their statements they confirmed a quake was possible but unlikely. So what we have to ask is this: based on the data they had, was that conclusion faulty?

To answer that you only need to ask – do all tremors lead to quakes? Well no, most don’t. So the quake was by no means likely, let alone inevitable. They were not covering up. None in the group was suppressed or censored. I can only conclude that they, after years of chasing tremors, had come to discount the value of tremors as indicators.

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And today, a judge decided that their advice was not only wrong, but criminally wrong.

The problem here is that science is not a PR exercise. If scientists put spin on data, they lose credibility. If they cry wolf, they lose credibility. The only safe way to do science is to stick to the facts. The facts did not indicate an imminent earthquake. The judge does not seem to realise: scientists cannot, and do not claim to be able to, predict earthquakes.

Perhaps the judge should read Taleb’s treatise on rare events, and he’d seen that just because something is unlikely (as the scientists said) does not protect us from the possibility it may still happen.

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So what we have seen today is this: scientists, giving their edified analysis, have been thrown in jail. The mob are satisfied, but you should be distinctly worried.